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A Dynamic Level-k Model in Sequential Games
Teck-Hua Ho
Xuanming Su
University of Pennsylvania
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Ho, T., & Su, X. (2013). A Dynamic Level-k Model in Sequential Games. Management Science, 59 (2), 452-469.http://dx.doi.org/ 10.1287/mnsc.1120.1645
A Dynamic Level-k Model in Sequential Games
AbstractBackward induction is a widely accepted principle for predicting behavior in sequential games. In the classic example of the “centipede game,” however, players frequently violate this principle. An alternative is a “dynamic level-k” model, where players choose a rule from a rule hierarchy. The rule hierarchy is iteratively defined such that the level-k rule is a best response to the level-(k-1) rule, and the level-∞ rule corresponds to backward induction. Players choose rules based on their best guesses of others' rules and use historical plays to improve their guesses. The model captures two systematic violations of backward induction in centipede games, limited induction and repetition unraveling. Because the dynamic level-k model always converges to backward induction over repetition, the former can be considered to be a tracing procedure for the latter. We also examine the generalizability of the dynamic level-k model by applying it to explain systematic violations of backward induction in sequential bargaining games. We show that the same model is capable of capturing these violations in two separate bargaining experiments.
Keywords
level-k models, learning, sequential games, backward induction, behavioral game theory
Disciplines
Finance and Financial Management | Social and Behavioral Sciences
A Dynamic Level-k Model in Games
Teck-Hua Ho and Xuanming Su∗
September 21, 2010
Backward induction is the most widely accepted principle for predicting behav-ior in dynamic games. In experiments, however, players frequently violate this principle. An alternative is a 2-parameter “dynamic level-k” model, where play-ers choose a rule from a rule hierarchy. The rule hierarchy is iteratively defined such that the k rule is a best-response to the (k − 1) rule and the level-∞ rule corresponds to backward induction. Players choose rules based on their best guesses of others’ rules and use past plays to improve their guesses. The model captures two systematic violations of backward induction, namely limited induction and time unraveling, and helps to resolve paradoxical behaviors in the centipede game, finitely repeated prisoner’s dilemma, and chain store game, three canonical games where backward induction performs poorly. The dynamic level-k model can be considered as a tracing procedure for backward induction because the former converges to the latter in the limit.
Keywords: Level-k Models, Learning, Dynamic Games, Backward Induction, Be-havioral Game Theory, Experimental Economics
∗Haas School of Business, University of California at Berkeley. Authors are listed in alphabetical order.
We thank Colin Camerer and Juin-Kuan Chong for collaboration in the project’s early stages, and Vince Crawford for extremely helpful comments. We are also grateful for constructive feedback from seminar par-ticipants at University of California at Berkeley, University of Pennsylvania, and University of Oxford. Direct correspondence to any of the authors. Ho and Su: Haas School of Business, University of California at Berke-ley, BerkeBerke-ley, CA 94720-1900, Email: Ho: [email protected]; Su: [email protected].
I.
Introduction
Players often interact with one another over multiple stages in many economic situations. Economists invoke the principle of backward induction to predict behavior in these games. Under backward induction, players reason backward, replace each subgame by its optimal payoff, always choose rationally at each subgame, and use this iterative process to determine a sequence of optimal actions. Each player follows this procedure betting on others doing the same. This “divide and conquer” algorithm simplifies the game analysis and generates a sharp prediction of game play at every subgame.
However, there are several canonical games showing that people often violate backward in-duction. One of these games is the so-called “centipede” game (Rosenthal, 1981) (See top panel of Figure 1). In this game, there are 2 players (A and B) and 4 decision stages. Players are endowed with an initial pot of $5. In Stage I, Player A has the property rights to the pot. She can choose either to end the game by taking 80% of the pot (and leaving the remaining 20% to Player B) or to allow the pot to double by passing the property rights to Player B. In Stage II, it is now Player B’s turn to make a similar decision. Player B must now decide whether to end the game by taking 80% of $10 or to let the pot double again by passing the property rights back to Player A. This social exchange process leads to large financial gains as long as both players surrender their property rights at each stage. At Stage IV, Player B can either take 80% of $40 (i.e., $32) or pass and be left with 20% of a pot of $80 (i.e., $16).
The principle of backward induction generates a sharp prediction for this game by starting the analysis in the very last stage. Player B should take at Stage IV because 80% of $40 is larger than 20% of $80. Anticipating this choice, Player A should take at Stage III since 80% of $20 is larger than 20% of $40. Continuing with this line of logic, backward induction makes a surprising prediction: Player A always takes immediately in Stage I and Outcome 4 occurs with probability 1 (i.e. Outcomes 0-3 should not occur). Moreover, the same prediction holds even if the game continues for more stages and with more dramatic potential gains. For example, the bottom panel of Figure 1 shows the same game with 6 stages and the same sharp prediction holds (i.e., Outcome 6 occurs with probability 1). Introspection suggests that this prediction is unlikely to occur. This is so because as long as the game proceeds to Stage III, both players would have earned more money than the backward induction outcome.
Figure 1: 4-stage (top) and 6-stage (bottom) centipede games.
Indeed, very few subjects (about 6%) choose to take immediately in experimental centipede games conducted by McKelvey and Palfrey (1992). Many subjects choose to take in the intermediate stages, approximately half way through the games (i.e., leading to Outcomes 2-3 in 4-stage and Outcomes 3-4 in 6-stage games). Clearly, this pattern of behavior runs counter to backward induction. Moreover, the observed behavior frequently leads to higher cash earnings for all subjects. Other well-known games where backward induction performs poorly include the repeated prisoner’s dilemma (Axelrod, 1984) and the chain store market entry game (Selten, 1978). In the latter two cases, the unique prediction prescribed by back-ward induction was rarely played especially when subjects were inexperienced.
There are two stylized facts concerning the violations of backward induction. First, play-ers violate backward induction less in a game with fewer subgames (or stages). That is, players’ behaviors deviate less from backward induction in simpler games. For instance, we observe fewer violations of backward induction in 4-stage than in 6-stage game (see Figure 1). We call this behavioral tendency limited induction. Second, players unravel over time
as they play the same game repeatedly over multiple rounds. That is, players’ behaviors converge towards backward induction over time. For instance, we observe fewer violations of backward induction in the last round than in the first round of the experimental centipede games in Figure 1. This behavioral tendency is termed time unraveling. The inability of backward induction to account for the two empirical stylized facts poses modeling challenges for economists.
In this paper, we propose an alternative to backward induction, a “dynamic level-k” model, that generalizes backward induction and accounts for limited induction and time unraveling. In the dynamic level-k model, players choose a level-k rule, Lk (k = 0, 1, 2, 3 . . .), from a set
of iteratively defined rules and the chosen rule prescribes an action at each subgame (Stahl and Wilson, 1995; Stahl, 1996; Ho et al. 1998; Costa-Gomes et al. 2001; Costa-Gomes and Crawford, 2006; Crawford and Iriberri, 2007a, 2007b). The rule hierarchy is defined such that the level-k rule best-responds to the level-(k − 1) rule and the level-∞ rule corresponds to backward induction. Since players choose a rule based on their beliefs of others’ rules, they essentially are subjective expected utility maximizers.
Players are heterogenous in that they have different initial guesses of others’ rules and con-sequently choose different initial rules. The distribution of the initial guesses is assumed to follow a Poisson distribution. These initial guesses are updated according to Bayes’ rule based on game history (cf. Stahl, 2000). Consequently, players develop more accurate guess of others’ rules and may choose different rules over time.
We prove that the dynamic level-k model can account for limited induction and time unravel-ing properties in the centipede game, the finitely repeated prisoner’s dilemma, and the chain store game. Consequently our model can explain passing in the centipede game, cooperation in the repeated prisoner’s dilemma, and fighting by the incumbent in the chain-store game. All these behaviors are considered paradoxical under backward induction but are predicted by the dynamic level-k model. In addition, the dynamic level-k model is able to capture the empirical stylized fact that behavior will eventually converge to backward induction over time, and hence the former can be considered as a tracing procedure for the latter.
(1992) and find that our model fits the data significantly better than backward induction and static level-k model. We rule out two alternative explanations including the reputation-based model of Kreps et al. (1982) and a model allowing for social preferences (Fehr and Schmidt, 1999). Overall, it appears that the dynamic level-k model can be an empirical alternative to backward induction.
The rest of the paper is organized as follows. Section II discusses the backward induction principle and its violations. Section III formulates the dynamic level-k model and applies it to explain paradoxical behaviors in the centipede game, iterated prisoner’s dilemma, and chain store game. Section IV fits the dynamic level-k model to data from experimental centipede game and rules out two alternative explanations. Section V concludes.
II.
Violations of Backward Induction
Backward induction uses an iterative process to determine an optimal action at each sub-game. The predictive success of this iterative reasoning process hinges on players’ complete confidence in others applying the same logic in arriving at the backward induction outcomes (Aumann, 1995). If players have doubts about others applying backward induction, it may be in their best interest to deviate from the prescription of backward induction. Indeed, subjects do and profitably so in many experiments.
If a player i chooses a behavioral rule Li that is different from backward induction (L∞),
one would like to develop a formal measure to quantify this deviation. Consider a game G with S subgames. We can define the deviation for a set of behavioral rules Li(i = 1, . . . I), one for each player, in game G as:
δ(L1, . . . , LI, G) = 1 S S X s=1 " 1 Ns Ns X i=1 Ds(Li, L∞) # , (II.1)
where Ds(Li, L∞) is 1 if player i chooses an action at subgame s that is different from the
prescription of backward induction and 0 otherwise, and Ns is the number of players who
are active at subgame s. Note that the measure varies from 0 to 1, where 0 indicates that players’ actions perfectly match the predictions of backward induction and 1 indicates that
none of the players’ actions agree with the predictions of backward induction.
Let us illustrate the deviation measure using a 4-stage centipede game. Let the behavioral rules adopted by player A and B be LA = {P, −, T, −} and LB = {−, P, −, T }
respec-tively (that is, player A will pass in Stage I and take in Stage III, and player B will pass in Stage II and take in Stage IV). Then the game will end in Stage III (i.e., Outcome 2). The deviation will be δ(LA, LB, G) = 1
4[1 + 1 + 0 + 0] = 1
2. Similarly, if L
A = {P, −, T, −}
and LB = {−, T, −, T }, then the game will end in Stage II (i.e., Outcome 3). This gives δ(LA, LB, G) = 1
4[1 + 0 + 0 + 0] = 1
4, which is smaller. Note that the latter behavioral rules
are closer to backward induction than the former behavioral rules.
Using the above deviation measure, we can formally state the two systematic violations of backward induction as follows:
1. Limited Induction: Consider two games G and G0 where G0 is a proper subgame of G. The deviation from backward induction is equal or larger in G than in G0. That is, the deviation from backward induction increases in S. Formally, for a group of players who adopt the same set of behavioral rules (Li, i = 1, . . . , I) in games G and G0, we have δ(L1, . . . , LI, G) ≥ δ(L1, . . . , LI, G0). Consequently, a good model must predict a
larger deviation in G than in G0 to be behaviorally plausible.
2. Time Unraveling: If a game G is played repeatedly over time, the deviation from back-ward induction at time t converges to zero as t → ∞. That is, time unraveling implies δ(L1(t), . . . , LI(t), G) → 0 as t → ∞. Therefore, game outcomes will eventually be
consistent with backward induction after sufficiently many repetitions.
Let us illustrate limited induction and time unraveling using data from McKelvey and Palfrey (1992). These authors conducted an experiment to study behavior in 4-stage and 6-stage centipede games. Each subject was assigned to one of these games and played the same game in the same role 9 or 10 times. For each observed outcome in a game play, we can
compute the deviation from backward induction using Equation (II.1).1
Figure 2: Deviations from backward induction in the 4-stage and 6-stage centipede games (Data from McKelvey and Palfrey, 1992).
Figure 2 plots the cumulative distributions of deviations from backward induction in the 4-stage and 6-stage games respectively. The thick line corresponds to the 4-stage game and the thin line corresponds to the 6-stage game. The curve for the 6-stage game generally lies to the right of the curve for the 4-stage game except for high deviation values. A simple t-test shows that there is a statistically significant difference between the mean deviations in the two games. These results suggest that the limited induction property holds in this data set.
Figure 3 plots the cumulative distributions of deviations from backward induction in the first and the final round of the 4-stage game. The thick line corresponds to the first round and the thin line corresponds to the final round of game plays (similar results occur for the 1Since subjects did not indicate what they would have chosen in every stage (i.e., data were not collected
using the strategy method), we do not observe what the subjects would have done in subsequent stages if the game ended in an earlier stage. In computing the deviation, we assume that subjects always choose to take in stages beyond where the game ends. Therefore, the derived measure is a conservative estimate of the deviation from backward induction.
Figure 3: Deviations from backward induction in the first and the last round of the 4-stage centipede game (Data from McKelvey and Palfrey, 1992).
6-stage game). As shown, the curve for the first round lies to the right of the curve for the final round. A simple t-test shows that there is a statistically significant difference between the mean deviations in the first and last rounds. These results suggest that the deviation from backward induction decreases over time.
We shall use the deviation measure to establish the main theoretical results below. Specif-ically, we shall show that the deviation under dynamic level-k model is smaller in simpler games (i.e., limited induction property holds) and converges to zero over time (i.e., satisfies the time unraveling property).
III.
Dynamic Level-k Model
We consider a 2-player game that has S subgames. Players are indexed by i (i = 1, 2) and subgames by s (s = 1, . . . , S). Players are assumed to adopt a rule which prescribes an action at each subgame s. For example, in the centipede game studied by McKelvey and Palfrey (1992), S is either 4 or 6 and a rule Player A adopts in a 4-stage game can be LA= {P, −, T, −}, which specifies that the player will pass in Stage I and take in Stage III.
Players choose a rule from a rule hierarchy. Rules are denoted by Lk (k = 0, 1, 2, 3, . . .)
and are generated from iterative best-responses. In general, Lk is a best-response to Lk−1
at every subgame, and L∞ corresponds to backward induction. Under this rule hierarchy,
the deviation from backward induction (see equation (I.1)) is smaller if player i adopts a higher level rule (while others keep their rule at the same level). In fact, ceteris paribus, the deviation from backward induction is monotonically decreasing in k. In other words, the level of a player’s rule captures its closeness to backward induction.
Note that Lk prescribes the same behavior as backward induction in any game with k or
fewer subgames. In this regard, Lk can be viewed as a limited backward induction rule that
only works for simpler games. Putting it this way, a higher level rule is more likely to coin-cide with backward induction in a wider class of games so that the level of a rule measures its degree of resemblance to backward induction.
If every player is certain about others’ rationality, all players will choose L∞. However, if
players have doubts about others’ rationality, it is not in their best interest to apply back-ward induction. Instead, they should form beliefs over rules adopted by others2 and choose a
best-response rule in order to maximize their expected payoffs. Therefore, under the model, players are subjective expected utility maximizers. In this regard, the dynamic level-k model is similar to the notion of rationalizable strategic behavior (Bernheim, 1984; Pearce, 1984; Reny, 1992) except that players always choose a rule from the defined rule hierarchy.
In a typical laboratory experiment, players frequently play the same game repeatedly. After each game round, players observe the rules used by their opponents and update their beliefs by tracking the frequencies of rules played by opponents in the past. Let player i’s rule counts at the end of round t be Ni(t) = (Ni
0(t), . . . , NSi(t)) where Nki(t) is the cumulative count of
rule Lk that has been used by opponents at the end of round t (Camerer and Ho, 1999; Ho et
2Subjects’ beliefs may depend on their knowledge of their opponents’ rationality. Players who play against
opponents who are known to be sophisticated will adopt a higher level rule. For example, Palacios-Huerta and Volijc (forthcoming) show that many chess players in the centipede game choose to pass when playing against student subjects but they choose to take immediately when playing against other equally sophisticated chess players. Levitt et al. (2009) however find chess players choose to pass when they play against each other. This discrepancy could be due to the difference in players’ perception of their opponents’ rationality.
al., 2007). Note that for a game with S subgames, all rules of level S or higher will prescribe the same action at each subgame and hence we pool them together and collectively call them LS. Given these rule counts, player i forms a belief Bi(t) = (B0i(t), . . . , BiS(t)) where
Bki(t) = N i k(t) PS k0=0Nki0(t) . (III.1) Bi
k(t) is player i’s belief of the probability that her opponent will play Lk in round t + 1.
The updating equation of the cumulative count at the end of round t is given by:
Nki(t) = Nki(t − 1) + I(k, t) · 1, ∀k (III.2) where I(k, t) = 1 if player i’s opponent adopts rule Lkin round t and 0 otherwise. Therefore,
players update their beliefs based on the history of game plays. This updating process is consistent with Bayesian updating involving a multinomial distribution with a Dirichlet prior (Fudenberg and Levine, 1998, Camerer and Ho, 1999). As a consequence of the updating process, players may adopt a different best-response rule in round t + 1.3
3The above updating rule assumes that subjects observe rules chosen by opponents. This is possible if
the strategy method is used to elicit subjects’ contingent action at each subgame. When the opponents’ chosen rules are not observed, the updating process is still a good approximation because subjects may have a good guess of their opponents’ chosen rules in most simple games (e.g., centipede games). More generally, the updating of Ni
k(t) depends on whether player i adopts a higher or lower level rule than her opponent.
If the opponent uses a higher level rule (e.g., the opponent takes before the player in the centipede game), then we have similar to the above:
Nki(t) = Nki(t − 1) + I(k, t) · 1 (III.3) where I(k, t) = 1 if opponent adopts an action that is consistent with Lk in round t and 0 otherwise. If
player i adopts a higher level rule k∗(e.g., takes before the opponent in the centipede game), the player can only infer that the opponent has chosen some rule that is below k∗. Then we have:
Nki(t) = Nki(t − 1) + I(k ≤ k∗) · N i k(t − 1) Pk∗ k0=0Nki0(t − 1) (III.4) where I(k ≤ k∗) = 1 if k ≤ k∗ and 0 otherwise. This updating process assigns a belief weight to all lower level rules that are consistent with the observed outcome. The weight assigned to each consistent rule is proportional to its prior belief weight. For this alternative updating process, the main results for the centipede game (i.e., Theorems 1 and 2) still go through. However, we are not able to prove the same results for the repeated prisoner’s dilemma and the chain store game.
Player i chooses the optimal rule Lk∗ in round t + 1 from the rule hierarchy {L0, L1, . . . , LS}
based on belief Bi(t) in order to maximize expected payoffs. Let a
ks be the specified action
of rule Lk at subgame s. Player i believes that action ak0s will be chosen with probability
Bki0(t) by the opponent. Hence, the optimal rule chosen by player i is:
k∗ = argmaxk=1,...,S S X s=1 ( S X k0=1 Bki0(t) · πi(aks, ak0s) ) , (III.5)
where πi(aks, ak0s) is player i’s payoff at subgame s if i chooses rule Lk and the opponent
chooses Lk0 rule (cf. Camerer et al., 2004).
Note that we model learning across game rounds but not across stages within a game round. This is clearly an approximation. For a general game, a player could potentially update her belief about the opponent’ rule within a round as the game unfolds. Specifically, a player may rule out a potential rule used by her opponent if an observed choice by the opponent at a particular stage is inconsistent with that rule. For example, in the 4-stage centipede game, if player B observes player A passing in stage I, player B will infer that player A’s chosen rule must be level 3 or lower. Nevertheless, we believe that our model is a good starting point for 2 reasons. First, in the 3 games we study below (centipede game, repeated prisoner’s dilemma, and chain-store game), the posterior belief at the end of a game round remains the same whether or not within-round learning is modeled explicitly. This is so because no new information is revealed once a player’s opponent chooses an action that is contrary to the prior belief. For instance, in the centipede game, when a player is surprised by an opponent who takes earlier than expected, the game ends immediately and no additional information is revealed.4 Second, within-round learning frequently generates prediction that is contrary
to observed behavior. For example, in the centipede game, the second player who expects the first player to take immediately will be surprised if the latter passes. If we incorporate within-round learning, this will lead the second player to put more weights on the lower level rules. As a consequence, the second player will be more likely to pass, which generally runs 4Similarly, in the repeated Prisoner’s dilemma, players’ subsequent actions after a defection by some
player does not contain new information as long as players choose from the rule hierarchy specified below. In the chain store game, once the incumbent shares the market, subsequent actions by entrants will provide no new information as long as they choose from the defined rule hierarchy.
counter to the observed data.
We need to determine player i’s initial belief Bi(0). We define Ni(0) such that Ni
k(0) = β
for some k and 0 otherwise, where the parameter β captures the weight assigned to initial belief. In other words, player i places all the initial weight on a particular rule Lk and zero
weight on all other rules. Different players have different guesses about others’ rationality and hence place the initial weight on a different k. A player who places the initial weight on k will choose an initial rule of Lk+1 in round 1. The heterogeneity of players’ initial beliefs
is captured by a Poisson distribution. The proportion of players who hold initial belief k is given by:
φ(k) = e
−λ· λk
k! ; k = 0, . . . , S. (III.6)
For example, a φ(0) proportion of players believe that their opponents will play L0 and thus
choose L1. Similarly, a φ(k) proportion of players believe that their opponents will play Lk
and best-respond with Lk+1. All players best-respond given their beliefs in order to maximize
their expected payoffs. Note that no players will choose L0 under our model. That is, L0
only occurs in the minds of the higher level players.5
The dynamic level-k model, similar to the cognitive hierarchy (CH) model, uses a Poisson distribution to capture player heterogeneity. The single parameter Poisson approach allows these models to be tractably used as a building block for theoretical analysis. However, the parsimony of a parametric approach comes with an empirical limitation. Specifically, the Poisson distribution often implies a non-negligible proportion of L0 players in empirical
applications of the CH model (see Camerer et al., 2004). In contrast, using a general discrete distribution, Costa-Gomes and Crawford (2006) and Crawford and Iriberri (2007a, 2007b) show that the estimated proportion of L0 players is frequently zero. In agreement with
the above observation, our proposed dynamic level-k model assumes that there are no L0
players. Unlike the CH model, which captures heterogeneity in players’ rules, the dynamic level-k model captures heterogeneity in players’ beliefs of others’ rules and allows players to best-respond to their beliefs. For example, φ(0) represents the proportion of L0 players in
5In centipede games, this implies that the dynamic level-k does not admit any occurrence of Outcome 0,
where players pass all the way. No model with only self-interested players can account for such behavior. Our model can easily be extended by incorporating social preferences to capture this kind of phenomenon.
the CH model, but it represents, in our model, the proportion of players who believe that their opponents play L0 and thus choose L1. Therefore, our proposed dynamic level-k model
excludes L0 players.6 In this way, the dynamic level-k model retains the parsimony of the
Poisson distribution while improving its empirical validity.
The dynamic level-k model is different from the static level-k and cognitive hierarchy mod-els in 3 fundamental aspects. First, players in our model are not endowed with a specific thinking type. That is, players in our model are cognitively capable of choosing any rule but always choose the level that maximizes their expected payoff. In other words, players in our model are not constrained by reasoning ability. Second, players in our model may be aware of others who adopt higher level rules than themselves. In other words, a player who chooses Lk may recognize that there are others who choose Lk+1 or higher but still prefers
to choose Lk because there is a large majority of players who are Lk−1 or below. On the
other hand, the static level-k and CH models assume that players always believe they are the highest level thinkers (i.e., the opponents are always of a lower level rule). Third, unlike the static level-k and CH models, players in the dynamic level-k model may change their rules as they collect more information and update beliefs about others. Specifically, a player who interacts with opponents of higher level rules may advance to a higher rule. Similarly, a players who interacts with opponents of lower level rules may switch to a lower level rule in order to maximize their expected payoffs.
In summary, the dynamic level-k model has 2 parameters, β and λ. The parameter λ captures the degree of heterogeneity in initial beliefs and the parameter β captures the strength of the initial belief, which in turn determines players’ sensitivity to game history. The 2-parameter dynamic level-k model nests several well-known special cases. When λ = ∞, it reduces to backward induction. If β = ∞, players have a stubborn prior and never respond to game history. This reduces our model to a variant of the static level-k model.7 Consequently, we
can empirically test whether these special cases are good approximations of behavior using 6It is possible to generalize this setup by introducing a segment of non-strategic players (level 0). Let
π be the proportion of non-strategic players and 1 − π be the proportion of strategic players (level 1 or higher). As described above, the Poisson model can be used to describe the distribution of strategic players. Consequently, the dynamic level-k model is simply a special case with π = 0.
7Note that under the dynamic level-k model, while players may believe others are level-0, they themselves
the standard generalized likelihood principle.
Below, we apply the dynamic level-k model to explain violations of backward induction in three canonical games (centipede game, finitely repeated prisoner’s dilemma, and chain store game). In each case, we prove that the dynamic level-k model can account for the limited induction and time unraveling properties of the data.8
A.
Centipede game
McKelvey and Palfrey (1992) study centipede games with an even number of stages (e.g., 4 and 6 stages). Hence, we focus on this class of games (ie., S is even). The above au-thors also assume there is a proportion of players who always pass in every stage. To avoid over-fitting, we assume that L0 corresponds to the same strategy. That is, we choose
L0 = {P1, P2, . . . , PS−1, PS}. (Note that odd-numbered components apply to Player A and
even-numbered components apply to Player B.) A player who believes her opponent uses L0 will maximize her payoff by adopting L1 = {P1, P2, . . . , PS−1, TS}. In general, Lk
best-responds to Lk−1 so that Lk = {P1, . . . , PS−k, TS−k+1, . . . , TS}. Therefore in a centipede
game with S stages, LS = LS+1 = . . . = L∞. Put differently, all rules LS or higher prescribe
the same action at each stage as L∞ (the backward induction rule). Consequently, we pool
8Besides these three games, alternating-offers bargaining game is also frequently used to show systematic
violations of backward induction. Camerer et al. (1993) is one of the earliest experiments to show that subjects rarely look at some crucial payoff information that is needed to apply backward induction. Johnson et al. (2002) builds on this work by showing that these violations cannot be explained by social preferences since subjects make similar offers to robot players. In a related study, Binmore et al. (2002) identify 2 behavioral properties implied by the principle of backward induction and show that subjects frequently violate them. The first behavioral property is called truncation consistency, which suggests that behavior should be invariant to replacing a subgame with its equilibrium payoff. In the centipede game, this implies that behaviors in a 4-stage game should be identical to behavior in the first four stages in a 6-stage game. The experimental data from McKelvey and Palfrey (1992) strongly rejected this property. The limited induction property predicts this violation and hence can help explain why truncation consistency is violated in alternating-offers bargaining games. The second behavioral property identified is called subgame consistency, which implies that behavior should be invariant to whether the same game is played independently or part of a larger game. The dynamic level-k model cannot explain violation of subgame consistency in alternating-offers bargaining games. There was, however, no evidence to suggest that subgame consistency is violated in experimental centipede games (McKelvey and Palfrey, 1992).
all these higher level rules together and collectively call them LS.
Under the dynamic level-k belief model, players choose rules from the rule hierarchy Lk
(k = 0, 1, 2, 3, . . .). Let Li(t) be the rule of Player i (where i = A, B) in time t. Then
δ(LA(0), LB(0), G) is the deviation from backward induction L∞ in game G at time 0. For
example, if Players A and B both play L2, the deviation is 12 in a 4-stage and 23 in a 6-stage
centipede game. Let G4 and G6 be the 4-stage and 6-stage games respectively. Then we
have the following theorem:
Theorem 1 In the centipede game, the dynamic level-k model implies that the limited in-duction property is satisfied. Formally, if the initial distribution of beliefs is the same (i.e., λ is the same) in both games GS and GS0 with S < S0, then the expected deviation from
backward induction in GS, δ(LA(0), LB(0), GS), is smaller than in GS0, δ(LA(0), LB(0), GS0).
Proof: See appendix.
Theorem 1 suggests that the dynamic level-k model gives rise to a smaller deviation from backward induction in a game with a smaller number of stages. This result is consistent with the data presented in Figure 2. The Appendix gives the detailed proof but the basic idea of the proof is outlined here. Given any rule combination LA(0) and LB(0) for the
players, let KA(0) = 2 · bLA2(0)c and KB(0) = 2 · dLB(0)
2 e − 1. The outcome is identical in
both games (counting from the last stage) (see Figure 1). Specifically, the game outcome is z = max{KA(0), KB(0)} and the number of actions that are inconsistent with backward
induction is S − z. As a consequence, G6 has a larger deviation than G4 (see equation (I.1)).
Since the initial distribution of beliefs is the same in both games (i.e., having the same λ), the initial expected deviation must be higher in G6.
Figure 3 shows that the expected deviation from backward induction becomes smaller over time. There is a question of whether this trend will persist and eventually unravel to back-ward induction outcome. The following theorem formally states that this is indeed the case for the dynamic level-k model.9
9Theorem 2 shows that all players will choose L
S = L∞as t → ∞. However, during the transient phase,
it is possible for players to switch from Lk to Lk0 (k0 < k) if they repeatedly encounter opponents of lower
Theorem 2 In a centipede game with S stages, the dynamic level-k model implies that the time unraveling property is satisfied. Formally, the deviation from backward induction δ(LA(t), LB(t), G) converges to zero and all players will choose L
S = L∞ as t → ∞. That
is, players will eventually take in every stage.
Proof: See appendix.
Theorem 2 states that the dynamic level-k model satisfies the time unraveling property. The basic idea of the proof is outlined here. No players choose L0. As a consequence, L1 players
will learn that other players are L1 or higher. Using our notation, this means that B0i(t)
will decline over time and the speed of decline depends on the initial belief weight β. For a specific β, there is a corresponding number of rounds after which all L1 players will move up
to L2 or higher. No players will then choose L0 and L1. In the same way, L2players will learn
that other players are L2 or higher and will eventually move to L3 or higher. Consequently,
we will see a ‘domino’ phenomenon whereby lower level players will successively disappear from the population. In this regard, players believe that others become more sophisticated over time and correspondingly do so themselves. In the limit, all players converge to LS (and
the learning process ceases).
The proof also reveals an interesting insight. The number of rounds it takes for Lk to
disap-pear is increasing in k. For example, when β = 1, it takes 6 rounds for L0 to disappear and
another 42 rounds for L1 to disappear. In general, for an initial belief weight β, it takes a
total of (7k− 1)β rounds for rule L
k to disappear. Note that each higher level rule takes an
exponentially longer time to be eliminated from the population.10 This result suggests that time unraveling occurs rather slowly in the centipede game. In the experiments conducted by McKelvey and Palfrey (1992), there are only 10 game rounds and no substantial learning is observed. In Section IV, we fit the dynamic level-k model and show that only L1 disappears
in their data set.
10In our model, the cumulative rule counts Ni
k(t) do not decay over time. If decay is allowed, that is,
Ni
k(t) = δ · N i
k(t − 1) + I(k, t) · 1, then unraveling can occur at a much faster rate. For example, if δ = 0, it
B.
Finitely Repeated Prisoner’s Dilemma
In the finitely repeated prisoner’s dilemma, backward induction prescribes that all players should defect in every repetition (see Figure 4 for a game with 10 repetitions). This is be-cause defection is a dominant strategy in every repetition. To explain cooperative behavior, Kreps et al. (1982) assume there is a proportion of players who always adopt the Tit-For-Tat (TFT) rule, which starts with cooperation and reciprocates by choosing the opponent’s action in the previous round. To avoid over-fitting, we choose L0 to be TFT. A player who
believes her opponent uses L0 will play TFT until the very last round when it is optimal to
defect. We denote this rule by L1 = {T F T, D}. Note that L1 is better than TFT because
defection yields a payoff of 5 in the last round while cooperation yields a payoff of 3; in contrast, {T F T, D, D} is inferior to {T F T, D} because the former yields a payoff of 5+1=6 in the last 2 rounds whereas the latter yields a payoff of 3+5=8 (see Figure 4). In general, Lk
best-responds to Lk−1 so that Lk= {T F T, k
z }| {
D, . . . , D}.11 Therefore, in a repeated prisoner’s
dilemma game with S repetitions, LS = LS+1 = . . . = L∞. Put differently, all rules LS or
higher prescribe the same actions as backward induction. Consequently, we pool all these higher level rules together and collectively call them LS.
Figure 4: Finitely repeated prisoner’s dilemma games (10 repetitions).
Theorem 3 In the finitely repeated prisoner’s dilemma, the dynamic level-k model implies 11Choosing cooperation initially followed by defection in the last k repetitions is also a best-response to
Lk−1. However, this simpler rule will perform poorly when played against a higher level rule. Since players
are fully aware of the entire rule hierarchy and may believe that others can use a higher level rule, it is in their best interest to adopt a more robust TFT based rule.
that the limited induction property is satisfied. Formally, if the initial distribution of beliefs is the same (i.e., λ is the same), then the expected deviation from backward induction is smaller in a game with a smaller number of repetitions.
Proof: See appendix.
The basic idea of the proof is similar to the one in the centipede game. For any combination of initial rules LA(0) and LB(0), let z = max{LA(0), LB(0)}. Then defection begins after
S − z repetitions, and both players will choose defection in the remaining z − 1 repetitions. As a consequence, a repeated prisoner’s dilemma with a larger number of repetitions will have a longer string of sustained cooperation before defection begins. Thus the expected deviation of backward induction is smaller for a game with fewer repetitions.
Theorem 4 In the finitely repeated prisoner’s dilemma with S repetitions, the dynamic level-k model implies that the time unraveling property is satisfied. Formally, the deviation from backward induction converges to zero and all players will choose LS = L∞ as t → ∞.
That is, players will eventually defect in every repetition.
Proof: See appendix.
Similar to before, no players choose L0 (i.e., TFT), so all players will defect in the last stage.
L1 players will learn over time that other players are L1 or higher and eventually move to
L2 or higher (i.e., L1 disappears). After this transition, all players will defect in the second
last repetition. The same logic continues and eventually all players will defect in every single repetition. Like before, the number of rounds for Lk to disappear is increasing in k. In
general, for an initial belief weight β, it takes a total of (2k − 1)β rounds for rule L k to
disappear. Note that the rate of convergence is faster for repeated prisoner’s dilemma than centipede game. In general, the rate of convergence depends on the payoff structure of the game.
C.
Chain Store Paradox
In the chain store game, an incumbent faces potential competition from numerous entrants in separate locations and must interact with each one of them sequentially (Selten, 1978).
Entrants in each location can either choose to enter the market or stay out. In Figure 5, there are 10 entrants. If an entrant chooses to stay out, the chain store receives a payoff of $5 m and the entrant receives a payoff of $1 m. If the entrant enters the market, the chain store must then choose to either stage a price war or share the market with the entrant. If the chain store fights, both players receive $0 m. If the chain store shares, both players receive $2 m.
Figure 5: Chain store game with 10 entrants over 10 interactions.
Focusing on the very last entrant, the chain store will choose to share the market with the entrant if it enters. Repeating this logic, backward induction predicts that each entrant will enter at its respective location and the chain store will share the market with each one of them. This seems implausible because introspection suggests that the chain store is likely to fight at least in some of the markets and not all entrants will choose to enter in their locations. Specifically, Selten (1978) wrote:
The disturbing disagreement between plausible game behavior and game theoretical reasoning constitutes the chain store paradox.
To explain entry and sharing, Jung et al. (1994) and Camerer et al. (2002) assume that there is a probability that the incumbent is a fighter type, who always fights upon entry. To avoid over-fitting, we assume that the LI
0 incumbent will always fight conditional on
en-trance. We assume that LE
1 entrants who believe the incumbent uses LI0 will maximize their
payoff by adopting the so-called grim trigger rule (GTR). The GTR rule prescribes that an entrant contemplating entry will choose to stay out unless the chain store is observed to have shared the market in the past. Given the sequential nature of the game, LI
1 incumbent
will best-respond to LE1 entrants by always fighting until the very last interaction when it is optimal to share the market condition on entrance, i.e., {F, F, . . . , F, S}.
In general, LEk entrants best-respond to a LIk−1 incumbent and this implies that LEk = {GT R,
k−1
z }| {
E, . . . , E}. Similarly, a LI
k incumbent best-responds to LEk−1 entrants and this
im-plies that LIk = {F, F, F,
k
z }| {
S, . . . , S}. Therefore, in a chain store game with S interactions, LE
S+1 = . . . = LE∞ and LIS = LIS+1 = . . . = LI∞. Put differently, all rules with level S or
higher prescribe the same actions as the rule of backward induction. Consequently, we pool all these higher level rules together and collectively call them LE
S+1 and LIS.
Note that we have defined two separate rule hierarchies, one for each role, because the incum-bent and entrants have a different strategy space. Further, in this game, each interaction is a sequential game where the incumbent chooses only after observing the action of the entrant. Consequently, our rule hierarchies are defined in a zig-zag structure where LEk entrants best-respond to LI
k−1 incumbent and LIk incumbent always best-responds to LEk entrants. Note
that the zig-zag structure is for notational convenience and it does not change the main results.
Theorem 5 In the chain store game, the dynamic level-k model implies that the limited induction property is satisfied. Formally, if the initial distribution of beliefs is the same (i.e., λ is the same) and all entrants have the same initial belief (i.e. they assign β to the same k), then the expected deviation from backward induction is smaller in a game with a smaller number of interactions.
In this theorem, the basic idea is similar to before. Let LI(0) and LE(0) be the initial rules
for incumbent and all entrants respectively. For any combination of rule LI(0) and LE(0), the outcome in the last z = max{LI(0), LE(0) − 1} interactions is the same regardless of the
total number of interactions. As a consequence, a chain store game with a larger number of repetitions will have a longer string of entrants staying out and the incumbent fighting before entrants begin to enter. Thus the expected deviation of backward induction is smaller for a game with fewer interactions.
Theorem 6 In the chain store game with S interactions, the dynamic level-k model im-plies that the time unraveling property is satisfied. Formally, the deviation from backward induction converges to zero, that is, the incumbent will choose LI
S = LI∞ and entrants will
choose LES+1 = LE∞ as t → ∞. That is, every entrant will eventually enter and incumbent will always share in every interaction.
Proof: See appendix.
The idea of the proof is similar to that in the previous two games. No incumbent will al-ways fight. Given this fact, LE
1 entrants will learn that the incumbent is LI1 or higher and
eventually will share in the last interaction. This implies that LE
1 will disappear. After this
happens, the LI1 incumbent will learn that entrants are LE2 or higher. As a consequence, LI1 will eventually disappear. The same logic continues and eventually all entrants will enter and the incumbent will share the market in every single interaction. This is consistent with backward induction.
D.
Properties of Level-0 Rule
The above rule hierarchies are generated from iterated best responses to a specific level-0 rule. To avoid over-fitting, we have used the level-0 rules from the existing literature. For instance, following McKelvey and Palfrey (1992), we use a decision rule that passes in every stage as level-0 in the centipede game. Similarly, as in Kreps et al. (1982), the Tit-For-Tat rule is used as level-0 in the repeated prisoner’s dilemma. Finally, in chain store game, we define always fighting as level-0 for the incumbent (following Jung et al., 1994 and Camerer
et al. 2002).
There are 2 ways to formally determine the level-0 rule. First, we can structurally estimate the level-0 rule using the data. One can parameterize the space of possible decision rules and empirically determine the best-fitting level-0 rule. Second, one can develop a theory for choosing a level-0 rule. We explore the latter approach below. Further research can provide a more comprehensive analysis.
A good level-0 rule may satisfy the following 2 attractive properties:
1. Maximize group payoff: A level-0 player always chooses a decision rule that if others do the same will lead to the largest total payoff for the group. For instance, in the centipede game, if every player chooses a decision rule that passes in every stage, the group will achieve the largest possible payoff.
2. Protect individual payoff: While maximizing group payoff, a level-0 player also ensures that the chosen decision rule is robust against continued exploitation by others. For instance, in the repeated prisoner’s dilemma, a level-0 who attempts to maximize group payoff by always cooperating may be exploited by an individual who always chooses to defect. A robust rule like Tit-For-Tat will prevent this kind of exploitation and promote cooperative behavior.
The above two properties do not define a unique level-0 rule in all games. However, they do narrow down the possible candidates for consideration. For the above 3 games that we consider, the chosen level-0 rules satisfy the two properties.
IV.
An Empirical Application to Centipede Game
A.
Dynamic Level-k Model
We use the dynamic level-k model to explain violations of backward induction in experimen-tal centipede games conducted by McKelvey and Palfrey (1992). The authors ran experi-ments using students subjects from Caltech (2 sessions) and Pasadena Community College (PCC) (4 sessions). In each subject pool, half the sessions were run on the 4-stage game
and the other half on the 6-stage game. Each experimental session consisted of 18 or 20 subjects and each subject played the game in the same role either 9 or 10 times. The ran-dom matching protocol was such that each player was matched with another player only once.
Contrary to backward induction, players did not always take immediately in both 4-stage and 6-stage games. In fact, a large majority passed in the first stage. For instance, 94% of the Caltech subjects passed in the first stage in 4-stage games (see Tables 1a-b). The distribution of game outcomes is unimodal with the mode occurring at the intermediate outcomes (Outcomes 2 and 3 in 4-stage games, Outcomes 3 and 4 in 6-stage games). These results present a considerable challenge to backward induction.
Tables 1a-b also suggest that Caltech subjects take one stage earlier than PCC subjects in both 4 and 6-stage games. Specifically, the modal outcome is Outcome 3 in 4-stage games and Outcome 4 in 6-stage games in Caltech subject pool while it is Outcome 2 in 4-stage games and Outcome 3 in 6-stage games in PCC subject pool. These results suggest that the two subject pools exhibit different levels of sophistication. We estimate the dynamic level-k model using the data from both subject pools with a goal of explaining their main features.
Outcome 4 3 2 1 0
Caltech (N =100) 0.06 0.43 0.28 0.14 0.09 PCC (N =181) 0.08 0.31 0.42 0.16 0.03
Table 1a: A Comparison of Outcomes in 4-stage Games Between Caltech and PCC subjects
Outcome 6 5 4 3 2 1 0
Caltech (N =100) 0.02 0.09 0.39 0.28 0.20 0.01 0.01
PCC (N =181) 0 0.05 0.09 0.44 0.28 0.12 0.02
Table 1b: A Comparison of Outcomes in 6-stage Games Between Caltech and PCC subjects
We conduct a conservative test for our model by specifying the model prediction for all game rounds without using the real-time data.12 As a consequence, the prediction of the dynamic
12It is possible to use an alternative approach where real-time data is used to adjust the model prediction.
level-k model for every game round is completely parameterized by λ and β. Conditional on λ and β, the model generates players’ choice probabilities for each rule Lk for all game
rounds. These choice probabilities in turn determine the distribution of outcomes over time. Let Pi
t(k|λ, β) be the model’s predicted probability that player i will choose a rule Lk in
round t. In the first round, P1i(k|λ, β) = φ(k − 1). In each subsequent round, to generate Pi
t(k|λ, β), we adopt a large sample approach by considering a large population of players.13
At the end of round t, players update their prior belief Bi(t − 1) based on their opponents’
chosen rule, which is drawn from Pti(k|λ, β) using a random matching protocol. This up-dating process gives rise to Bi(t). Each player then chooses a rule L
k that maximizes her
expected payoff. If two rules yield the same expected payoffs, players are assumed to always choose the simpler rule (i.e. a lower level rule).14 In this way, we can then compute the choice probabilities Pi
t+1(k|λ, β) in round t + 1.
Let O(t) be the outcome of a game in round t. For example, in the 6-stage game, O(t) = 6 if Player A takes immediately and O(t) = 0 if everyone passes to the end. Let P (O(t)) be the probability of observing outcome O(t). One can compute the probability of observing a data point O(t) conditional on λ and β. The probability is
P (O(t)) = " X k X k0 PtA(k|λ, β) · PtB(k0|λ, β) · I (H(k, k0), O(t)) # , (IV.1)
where I(H(k, k0), O(t)) is 1 if H(k, k0) = O(t) and 0 otherwise. We define H(k, k0) = max2 · bk
2c, 2 · d k0
2e − 1 .
The dynamic level-k model makes the following sharp predictions in the centipede game depending on the value of the β parameter:
provide a better fit but runs the risk of over-fitting. We tried this method and found that subjects moved up and down the rule hierarchy more frequently than the data would suggest. Consequently we chose not to adopt this approach.
13If we do not use the large sample approach, the model prediction for all game rounds can still be
generated. However, we can no longer assume that the distribution of choices is conditionally independent from the distribution of choices in previous rounds. Therefore, we need to consider all possible combinations of each player’s initial beliefs and how these beliefs evolve over time. This gives rise to a combinatorial problem as the number of rounds increases and makes the estimation computationally intractable.
14For example, in the centipede game, for player B, L
1 = L2 = {−, P, −, T }. We assume that player B
1. Players’ choice probabilities Pi
t(k|λ, β) do not change at every round. In fact, they
change at most twice within 10 game rounds.
2. For β ≥ 53, Pti(k|λ, β) = φ(k − 1) remains fixed over all 10 game rounds. For 53 > β ≥
1 6, P
i
t(k|λ, β) changes only once within 10 game rounds. For β < 1 6, P
i
t(k|λ, β) changes
twice within 10 game rounds.
3. The model starts by predicting that outcome 0 (i.e., players pass all the way) occurs with zero probability. At every change of the choice probabilities, the next higher outcome (outcome 1 at the first change and outcome 2 at the second change) will also occur with zero probability.
The dynamic level-k predicts that some game outcomes will occur with zero probability. To facilitate empirical estimation, we need to incorporate an error structure. To avoid specification bias, we use the simplest possible error structure (see Crawford and Iriberri 2007a). We assume a probability (1 − ) that the data matches our model prediction and a probability that the observed outcome is uniformly distributed over all possible outcomes. Hence, the likelihood function is given by:
L =Y t (1 − ) · P (O(t)) + · 1 S + 1 . (IV.2)
We fit the dynamic level-k model to the data using maximum likelihood estimation. We separately estimate the dynamic level-k model for Caltech and PCC subject pools because of the apparent differences in their behaviors (see Tables 1a-b). However, we use the same set of parameters to fit both the 4-stage and 6-stage games.
Table 2 shows the estimation results for Caltech subjects. As shown, the estimated param-eters are ˆλ = 1.32, ˆβ = 0.25, ˆ = 0.29, and the log-likelihood is −305.8. The estimated ˆ
λ = 1.32 suggests that initially, 27% of the players are level 1, 35% are level 2, 23% are level 3, and the remaining 15% are level 4 or higher. The estimated ˆβ lies between 16 and
5
3, suggesting that players’ choice probabilities change only once. The estimated ˆβ = 0.25
suggests a small initial belief weight, which implies that time unraveling can begin to occur very quickly and in fact after only 2 rounds.15
15Any value of β between 1 6 and
1
3 will predict a change after round 2 and we simply report the midpoint
Parameter Backward Induction Naive Belief Static Level k Full Model
λ ∞ 0 1.76 1.32
β ∞ 0.08 ∞ 0.25
1.00 0.97 0.21 0.29
Log Likelihood −357.1 −355.0 −312.3 −305.8
Table 2: Maximum Likelihood Estimates of Dynamic Level-k Model and Its Special Cases (Caltech Subjects)
Table 3a-b compare the model prediction and the actual data for the first 2 and last 8 rounds in the 4-stage and 6-stage games for Caltech subjects. The best fitted dynamic level-k model makes two predictions. First, in both games, Outcome 0 (i.e., both players pass all the way) should not occur in all rounds. Second, the model prescribes that unraveling occurs, and as a consequence Outcome 1 should not occur after round 2. These predictions were roughly consistent with the data in the following ways:
1. In both games, Outcome 0 occurs infrequently. For example, Outcome 0 occurs less than 1% of the time in 6-stage game. The model however fails to capture a nontrivial proportion of occurrence of Outcome 0 in the 4-stage game (even though the proportion declines from 15% to 8% after round 2).
2. In 4-stage game, the proportions of Outcomes 0 and 1 decline after 2 rounds. For example, the proportion of Outcome 1 decreases from 25% to 11%. In 6-stage game, Outcomes 0 and 1 rarely occur after round 2 (about 1% of the time).
Outcome 4 3 2 1 0
Rounds 1-2 Data (N =20) 0 0.35 0.25 0.25 0.15 Prediction ( = 0) 0.15 0.32 0.36 0.17 0 Rounds 3-10 Data (N =80) 0.08 0.45 0.29 0.11 0.08
Prediction ( = 0) 0.27 0.52 0.21 0 0
Table 3a: A Comparison of Data and Dynamic Level-k Model Prediction in 4-stage games (Caltech Subjects)
Outcome 6 5 4 3 2 1 0
Rounds 1-2 Data (N =20) 0 0 0.30 0.25 0.45 0 0
Prediction ( = 0) 0.01 0.04 0.13 0.29 0.36 0.17 0 Rounds 3-10 Data (N =80) 0.03 0.11 0.41 0.29 0.14 0.01 0.01
Prediction ( = 0) 0.01 0.06 0.24 0.47 0.21 0 0
Table 3b: A Comparison of Data and Dynamic Level-k Model Prediction in 6-stage games (Caltech Subjects)
Finally, the error rate is around 29% which is not negligible. We believe the value is at-tributed to the sharp prediction of the dynamic level-k model that both outcome 0 and 1 should not occur after round 2.
We also fit three nested cases of the dynamic level-k model. They are all rejected by the like-lihood ratio test. The backward induction model (i.e., λ = β = ∞) yields a log-likelike-lihood of −357.1, which is strongly rejected in favor of the full model (χ2 = 102.7). The naive
belief model (i.e., λ = 0) (which assumes that all players believe that their opponents will always pass) yields an estimated ˆβ = 0.08 and a log-likelihood of −355.0. The model is again strongly rejected with a χ2 = 98.5. The static level-k model corresponds to β = ∞.
Since players’ initial belief will persist throughout the game plays, they will always choose the same rule Lk across rounds. This restriction yields a parameter estimate of ˆλ = 1.76 and
provides a reasonable fit with a log-likelihood of −312.3. However, the static model is also rejected by the likelihood ratio test (χ2 = 13.1, p < 0.01).
Given the MLE estimates of ˆλ = 1.32, ˆβ = 0.25, we can generate the dynamic level-k model’s predicted frequencies for each of the outcomes. Figure 6 shows the actual and predicted fre-quencies of each outcome. The top panel shows the results for the 4-stage game and the bottom panel shows the results for the 6-stage game. Backward induction predicts that only Outcome 4 in 4-stage game and Outcome 6 in 6-stage game can occur (i.e., Player A takes immediately). This backward induction prediction is strongly rejected by the data. As shown, the dynamic-k model does a reasonable job in capturing the unimodal distribution of the outcomes. In addition, the dynamic level-k model is able to capture the two most frequently occurring outcomes. Specifically, the dynamic level-k model correctly predicts
that the two most frequently played outcomes are 2 and 3 in 4-stage game and 3 and 4 in 6-stage game.
Figure 6: Dynamic Level-k Model Fit (dark bars) and Data (light bars) (Caltech subjects).
We also fit our model to the data obtained from PCC subjects. Table 4 shows the estimation results. The dynamic level-k model yields parameter estimates of ˆλ = 1.30 and ˆβ = 1.67. This estimated initial belief weight ˆβ > 5
3 suggests that unraveling never occurs (i.e., there
is no change in the choice probabilities). Therefore, for PCC subjects, the dynamic level-k model delivers identical predictions as the static level-k model (i.e., β = ∞), which also yields an estimated ˆλ = 1.30. Both the static and dynamic level-k models give a log-likelihood score of −514.8. In contrast, the backward induction prediction (λ = β = ∞) gives a log-likelihood score of −646.5 and the naive belief model (λ = 0) gives a log-log-likelihood score of −644.3, so both models are strongly rejected by the likelihood ratio test.
Parameter Backward Induction Naive Belief Static Level k Full Model
λ ∞ 0 1.30 1.30
β ∞ 1.67 ∞ 1.67
1.00 0.99 0.08 0.08
Log Likelihood −646.5 −644.3 −514.7 −514.7
Table 4: Maximum Likelihood Estimates of Dynamic Level-k Model and Its Special Cases (PCC Subjects)
Since the dynamic level-k model predicts that there is no unraveling for PCC subjects, it is interesting to examine the actual data to see whether this is indeed the case. Tables 5a-b show the predictions of the dynamic level-k model as well as the actual data. The data suggests that PCC subjects exhibit a slower process of time unraveling compared to Caltech subjects (cf. Tables 3a-b). For example, after 2 rounds of play in the 6-stage game, Outcomes 0 and 1 still occur with more than 10% probability in PCC subjects while the same outcomes occur with only 2% chance in Caltech subjects. In fact, for both 4-stage and 6-stage games, the t-test provides evidence of a significant difference between the mean outcomes in the first 2 rounds and last 8 rounds for Caltech subjects. However, the same test is inconclusive for PCC subjects. These results show that within the 10 game rounds in the data, time unraveling has occurred for Caltech subjects but not for PCC subjects.
Outcome 4 3 2 1 0
Rounds 1-2 Data (N =38) 0.08 0.24 0.42 0.24 0.03 Prediction ( = 0) 0.26 0.39 0.27 0.08 0 Rounds 3-10 Data (N =143) 0.08 0.34 0.42 0.14 0.03
Prediction ( = 0) 0.26 0.39 0.27 0.08 0
Table 5a: A Comparison of Data and Dynamic Level-k Model Prediction in 4-stage games (PCC Subjects)
Outcome 6 5 4 3 2 1 0 Rounds 1-2 Data (N =38) 0 0.13 0.05 0.32 0.32 0.16 0.03
Prediction ( = 0) 0.03 0.10 0.20 0.31 0.27 0.08 0 Rounds 3-10 Data (N =143) 0 0.03 0.10 0.48 0.27 0.10 0.01
Prediction ( = 0) 0.03 0.10 0.20 0.31 0.27 0.08 0 Table 5b: A Comparison of Data and Dynamic Level-k Model Prediction
in 6-stage games (PCC Subjects)
Finally, given the MLE estimates of ˆλ = 1.30, ˆβ = 1.67 for PCC subjects, we can generate the dynamic level-k model’s predicted frequencies for each of the outcomes. Figure 7 shows the actual and predicted frequencies of each outcome. The top panel shows the results for the 4-stage game and the bottom panel shows the results for the 6-stage game. The plot shows that the dynamic level-k model provides a reasonably good fit of the data.
B.
Reputation-based Model
As indicated before, backward induction does not admit passing behavior in the centipede game. To explain why players pass, one can transform the game into a game of incomplete information by introducing some uncertainty over the player type (Kreps et al., 1982). If there is a small fraction θ of players that always pass (the so-called altruists), then even the self-interested players may find it in their best interest to pass initially. However, this reputation-based model generates the following empirical implications that run counter to the observed data (see McKelvey and Palfrey, 1992):
1. When θ is small (0 ≤ θ ≤ 491 ), the model will not correctly predict the most frequently occurring outcomes. For instance, the model predicts the modal outcome to be Out-come 4 in 4-stage game while the data suggests it should be OutOut-come 2. Similarly, the model predicts the two most frequently occurring outcomes to be Outcomes 4 and 5 in 6-stage game while the data suggests they should be Outcomes 2 and 3.
2. When θ is intermediate (491 < θ ≤ 17), the model will not admit Outcome 4 and 4-6 in 4-stage and 6-stage respectively.
3. When θ is large (θ > 17), the model predicts players either pass all way (Outcome 0) or take at the last stage (Outcome 1), which is contrary to the data.
The above discussion suggests that there is not a common θ value that will explain the unimodal distribution of game outcomes in both the 4-stage and 6-stage games well.
We estimate the reputation-based model using the centipede game data. Like before, we assume that there is a probability that the observed outcome will be evenly distributed over all possible outcomes. The best fitted model yields ˆθ = 0.050, ˆ = 0.62 for Caltech subjects and ˆθ = 0.075, ˆ = 0.31 for PCC subjects. The log-likelihood scores are −329.8 and −518.8 respectively, suggesting that the model did a worse job in fitting the data compared to the dynamic level-k model.
C.
Models of Social Preferences
Could the unimodal distribution of game outcomes in centipede game be attributed to social preferences? The answer is no. To see this we apply the inequity aversion model of Fehr
and Schmidt (1999). Let xi and x−i denote the material payoffs of player i and opponent −i
respectively. Then, the payoff of player i, Ui(xi, x−i), is given by:
Ui(xi, x−i) = xi− α · [xi− x−i]+− β · [x−i− xi]+ (IV.3)
where 0 < α < 1 captures a player’s aversion to being ahead and β > 0 captures a player’s aversion to being behind. The parameter α is between 0 and 1 because a player who is ahead will not give up a $1 to benefit her opponent less than $1. One can then solve the game by backward induction given these revised payoff functions.
Theorem 7 In the centipede game, social preferences lead to either taking immediately or passing all the way in both the 4-stage and 6-stage games. Specifically, social preferences pre-dicts that if 3·α−6·β < 2, players will take immediately; otherwise they will pass all the way.
Proof: See Appendix.
The above theorem states that a model of social preferences admits only either Outcome 0 or 4 in 4-stage game and either Outcome 0 or 6 in 6-stage game. The occurrence of either outcome depends on the fairness parameters α and β. If players are sufficiently averse to being behind (high β), they will take immediately, which is consistent to backward induc-tion. On the other hand, if players are sufficiently averse to being ahead (high α), they will pass all the way. Both predictions are inconsistent with the unimodal distribution of actual game outcomes.16
We estimate the social preference model using the centipede game data. Note that the model fit is identical to the backward induction model. If 3 · α − 6 · β ≤ 2, the model reduces to backward induction, so the best fit yields a log-likelihood of −357.1 for Caltech subjects and −646.5 for PCC subjects. If 3 · α − 6 · β ≥ 2, the model predicts that players pass all the way, so the best fit yields again a log-likelihood of −357.1 and −646.5. Hence the model performs poorly compared to the dynamic level-k model.
16Fey et al. (1996) study a constant-sum version of the centipede game in which there are initially two
equal piles of cash. Similar to the regular centipede game, each player takes turn to either take or pass. The player who takes will receive the larger pile of cash. Each time a player passes, a fraction of the smaller pile is transferred to the larger pile, so the division of the money becomes more unequal over stages. This design helps to control for social preferences. They show that about half the subjects still pass in the first stage and hence violate backward induction. This result suggests that violations of backward induction cannot be fully accounted for by social preferences.
V.
Conclusions
In economic experiments, backward induction is frequently violated. We develop a dynamic level-k model to explain two systematic violations of backward induction. First, players tend to deviate more from backward induction in games with a greater number of stages or sub-games. Our model captures this limited induction by allowing players to have heterogeneous initial beliefs about others’ rationality and hence adopt different rules from a rule hierarchy. Second, players move closer to backward induction over time. Our model captures this time unraveling by allowing players to update beliefs of their opponents’ rules and hence adjust their own rules over time. We show that this adjustment process leads to convergence to-wards backward induction.
We have applied the dynamic level-k model to 3 canonical games (centipede game, finitely repeated prisoner’s dilemma, and chain store game) that demonstrate the limitations of backward induction. In all three games, we prove that the limited induction and time un-raveling properties hold. Limited induction holds because the same rule always has a higher deviation from backward induction in a game with more stages. Time unraveling occurs because the dynamic level-k model implies a domino effect over time as lower level rules are successively eliminated.
We fit our model to experimental centipede games of McKelvey and Palfrey (1992). Our estimation results show that the dynamic level-k model captures the unimodal distribution of game outcomes in both 4-stage and 6-stage games reasonably well. Special cases including backward induction and the static level-k model are strongly rejected by the data. Interest-ingly, Caltech subjects, who are arguably more sophisticated, tend to learn faster than PCC subjects.
We rule out 2 alternative explanations for the observed phenomena in the centipede game. First, we show that the reputation-based model of Kreps et al. (1982) cannot capture the unimodal distribution of game outcomes in the manner consistent with the data. A limi-tation of the repulimi-tation-based model is that there is no common fraction of the altruistic players that will simultaneously account for the degree of passing in both the 4-stage and 6-stage games. Second, we show that social preferences in the form of inequity aversion (Fehr
